Dynamics on valuation spaces and applications to complex dynamics

Let f be a rational self-map
of the complex projective plane. A central problem when analyzing the
dynamics of f is to understand the sequence of degrees deg(f^n) of the
iterates of f. Knowing the growth rate and structure of this sequence in
many
cases enables one to construct invariant currents/measures for dynamical
system as well as bound its topological entropy. Unfortunately, the
structure of this sequence remains mysterious for general rational maps.
Over the last ten years, however, an approach
to the problem through studying dynamics on spaces of valuations has
proved fruitful. In this talk, I aim to discuss the link between
dynamics on valuation spaces and problems of degree/order growth in
complex dynamics, and discuss some of the positive results
that have come from its exploration.